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Fractal Dimension An important characteristic of a fractal The main tool for applications Self-similar fractals have a nice fractal dimension d given by N = (1/r) d where N is number of pieces, r is scaling factor, so d = ln N / ln r

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The Cantor Set Start with a unit interval, remove middle third interval, and continue to remove middle thirds of the subintervals Is self-similar and has a fractal dimension of ln 2/ ln 3

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Topology Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects Topology studies features of a space like connectivity or number of holes A topologist doesnt distinguish between a tea cup and a donut

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Homology Homology tries to distinguish between spaces by constructing algebraic and numerical invariants that reflect the connectivity of the spaces In general, the basic definitions are abstract and complicated For nice subsets of 2, the only non-trivial homology can be determined by counting holes

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Symmetric Binary Fractal Trees T(r, ) denotes tree with scaling ratio r (some real number between 0 and 1) and branching angle (real-valued angle between 0 º and 180 º ) Trunk splits into 2 branches, each with length r, one to the right with angle and the other to the left with angle Level k approximation tree has k iterations of branching

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Some Algebra A symmetric binary tree can be seen as a representation of the free monoid with two generators Two generator maps m R and m L that act on compact subsets Addresses are finite or infinite strings with each element either R or L

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Self-Contact For a given branching angle, there is a unique scaling ratio such that the corresponding symmetric binary tree is self-contacting. We denote this ratio by r sc. This ratio can be determined for any symmetric binary tree. If r < r sc, then the tree is self-avoiding. If r > r sc, then the tree is self-overlapping.

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Self-Contacting Trees The branching angles 90° and 135° are considered to be topological critical points, one reason being that the corresponding self-contacting trees are the only ones that are space-filling All other self-contacting trees have infinitely many generators for the first homology group

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Topology and Fractal Trees? At first, topology doesnt seem very useful for studying fractal trees- the topology is either trivial or too complicated Idea: study topological and geometrical aspects of a tree along with spaces derived from a tree What derived spaces?

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Closed ε-Neighbourhoods For a set X that is a subset of some metric space M with metric d, the closed ε- neighbourhood of X is X ε = { x | d(x, X) ε }

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Closed ε-Neighbourhoods of Trees The closed ε-neighbourhoods, as ε ranges over the non-negative real numbers, endow a tree with much additional interesting structure They are a function of r, θ, and ε What features do we study?

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Critical Values Critical set of ε-values for (r,θ) based on persistence Critical values of r for a given θ, based on complexity Critical values of θ, based on location Different relations give different classifications of the trees that focus on different aspects

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Specific Trees It is possible for a closed ε- neighbourhood to have infinitely many holes for non-zero value of ε T(r sc, 67.5°)

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Specific Trees It is often not straightforward to determine exact critical ε-values for a given tree, but they are not always necessary- sometimes estimates are good enough T(r sc, 120°)